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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubclsubN | Structured version Visualization version GIF version |
Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclsub.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
psubclsub.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
psubclsubN | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
2 | psubclsub.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
3 | 1, 2 | psubcli2N 39921 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) = 𝑋) |
4 | eqid 2734 | . . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
5 | 4, 1, 2 | psubcliN 39920 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) = 𝑋)) |
6 | 5 | simpld 494 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾)) |
7 | psubclsub.s | . . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾) | |
8 | 4, 7, 1 | polsubN 39889 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃‘𝐾)‘𝑋) ∈ 𝑆) |
9 | 6, 8 | syldan 591 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘𝑋) ∈ 𝑆) |
10 | 4, 7 | psubssat 39736 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ ((⊥𝑃‘𝐾)‘𝑋) ∈ 𝑆) → ((⊥𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) |
11 | 9, 10 | syldan 591 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) |
12 | 4, 7, 1 | polsubN 39889 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ((⊥𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) ∈ 𝑆) |
13 | 11, 12 | syldan 591 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) ∈ 𝑆) |
14 | 3, 13 | eqeltrrd 2839 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 ‘cfv 6562 Atomscatm 39244 HLchlt 39331 PSubSpcpsubsp 39478 ⊥𝑃cpolN 39884 PSubClcpscN 39916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-proset 18351 df-poset 18370 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-psubsp 39485 df-pmap 39486 df-polarityN 39885 df-psubclN 39917 |
This theorem is referenced by: pclfinclN 39932 |
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